ON FINITE SIMPLE AND NONSOLVABLE GROUPS ACTING ON CLOSED 4-MANIFOLDS

We show that the only finite nonabelian simple groups to admit a locally linear, homologically trivial action on a closed simply connected 4-manifold M (or on a 4-manifold with trivial first homology) are the alternating groups A(5), A(6) and the linear fractional group PSL(2, 7). (We note that for homologically nontrivial actions all finite groups occur.) The situation depends strongly on the second Betti number b(2)(M) of M and was known before if b(2)(M) is different from two, so the main new result concerns the case b(2)(M) = 2. We prove that the only simple group that occurs in this case is A(5), and then deduce a short list of finite nonsolvable groups which contains all candidates for actions of such groups.